Quasi-Rational Canonical Forms of a Matrix over a Number Field
|
|
- Rudolf Mason
- 5 years ago
- Views:
Transcription
1 Avace Lea Algeba & Matx Theoy, 08, 8, -0 ISSN Ole: ISSN Pt: X Qua-Ratoal Caocal om of a Matx ove a Numbe el Zhueg Wag *, Qg Wag, Na Q School of Mathematc a Stattc, Yacheg Teache Uvety, Yacheg, Cha How to cte th pape: Wag, Z.D., Wag, Q. a Q, N. (08) Qua-Ratoal Caocal om of a Matx ove a Numbe el. Avace Lea Algeba & Matx Theoy, 8, Receve: Octobe 8, 07 Accepte: Jauay 7, 08 Publhe: Jauay 0, 08 Abtact A matx mla to Joa caocal fom ove the complex fel a the atoal caocal fom ove a umbe fel, epectvely. I th pape, we futhe tuy the atoal caocal fom of a matx ove ay umbe fel. We ftly cu the elemetay vo of a matx ove a umbe fel. The, we gve the qua-atoal caocal fom of a matx by combg Joa a the atoal caocal fom. ally, we how that a matx mla to t qua-atoal caocal fom ove a umbe fel. Copyght 08 by autho a Scetfc Reeach Publhg Ic. Th wo lcee ue the Ceatve Commo Attbuto Iteatoal Lcee (CC BY 4.0). Ope Acce Keywo Matx, Joa Caocal om, Ratoal Caocal om, Qua-Ratoal Caocal om. Itoucto A matx mla to Joa caocal fom ove the complex fel a the atoal caocal fom ove a umbe fel, epectvely. Thu, Joa a the atoal caocal fom of a matx ove the complex fel ae mla. Recetly, Rajabalpou [] tue the ymmetzato of the Joa caocal fom, Abo et al. [] a Baoe et al. [3] cue the elato betwee the egetuctue a Joa caocal fom. Moeove, L [4] cue the popety of the atoal caocal fom of a matx, Lu [5] gave out a cotuctve poof of extece theoem fo atoal fom, a Rajabalpou [6] vetgate the atoal caocal fom va the plttg fel. I th pape, we futhe tuy the atoal caocal fom ove ay umbe fel. We ftly cu the cocept of elemetay vo of a matx ove ay umbe fel. The, we gve the qua-atoal caocal fom of a matx by combg Joa a the atoal caocal fom. ally, we how that a matx mla to t qua-atoal caocal fom ove ay umbe fel. DOI: 0.436/alamt Ja. 0, 08 Avace Lea Algeba & Matx Theoy
2 Z. D. Wag et al.. Joa a Ratoal Caocal om Gve a matx A, t a teetg wo to f a mple matx that mla to A. We ow that uch a mple matx Joa caocal fom o the atoal caocal fom of A. Lemma.. ([7], pp 44-47) A matx A mla ove the complex fel to Joa caocal fom of t, uch Joa caocal fom uque up to a eaagemet of the oe of t chaactetc value,.e., A mla to the qua-agoal matx of oe whee J J J = J c c 0 0 J = c m m calle a elemetay Joa matx wth chaactetc value m + m + + m =. a Defto.. o a o-cala moc polyomal ove a umbe fel P, the matx a 0 0 a B = 0 0 a 0 0 a c, =,,, λ = λ + aλ + + a calle the compao matx o obeu matx of the moc polyomal. A polyomal matx, o λ-matx, a ectagula matx A λ whoe elemet ae polyomal λ ( 0 j j j j ) A λ = a λ = a λ + a λ + + a. Hee the laget of the egee of the polyomal aj. Two polyomal matce A a B λ ae calle equvalet f oe of them ca be obtae fom the othe by mea of ome elemetay opeato. A abtay ectagula polyomal matx equvalet to a caocal matx a a a DOI: 0.436/alamt Avace Lea Algeba & Matx Theoy
3 Z. D. Wag et al. whee the polyomal a a a,,, ae ot etcally equal to zeo a each vble by the peceg oe. Let A be a polyomal matx of a,.e., the matx ha mo of oe ot etcally equal to zeo, but all the mo of oe geate tha ae etcally equal to zeo λ. We eote by D j the geatet commo vo of all the mo of oe j A ( j =,,, ). It eay to ee that the ee D, D,, D each polyomal vble by the peceg oe (ee [8], pp 39-40). A eay vefcato how mmeately that the elemetay opeato chage ethe the a of A o the polyomal D, D,, D. Thu, =, the coepog quotet wll be eote by D D λ = λ, λ =,, λ = D D vaat ue elemetay opeato a D = = = a, a,, a. The polyomal,,, of the λ-matx A. ae calle the vaat polyomal Defto.. ([8], pp 44-45) Let A= ( a j ) be a matx. We fom t chaactetc matx λ a a a a λ a a λe A =. a a λ a The chaactetc matx a λ -matx of a. It vaat polyomal D λ D λ λ λ =, =,, = D D D ae calle the vaat polyomal of the matx A. It eay to ee that the vaat polyomal of the compao matx B of the moc polyomal ae,,,. Defto.3. The followg qua-agoal matx B B C = B calle the ect um of the compao matce B of o-cala moc polyomal,,, uch that + ve fo =,, a a to be atoal caocal fom. The vaat polyomal of the atoal caocal fom matx B Defto.3 ae DOI: 0.436/alamt Avace Lea Algeba & Matx Theoy
4 Z. D. Wag et al.,,,,,,. Lemma.. ([7], pp 38-39, Theoem 5) A matx A mla ove a umbe fel P to oe a oly oe matx whch atoal caocal fom,.e., A mla to the qua-agoal matx B B C = B whee B, B,, B ae the compao matce of the o-cala vaat polyomal,,, of matx A. 3. The Elemetay Dvo of a Matx ove a Numbe el Let P be a umbe fel. The a o-cala moc polyomal P[ x ] ca be factoe a a pouct of moc eucble polyomal P[ x ] oe a, except fo oe, oly oe way. I the factozato of a gve o-cala moc polyomal f ( x ), ome of the moc eucble facto may be epeate. If p ( x) p ( x) p ( x),,, ae the tct moc eucble polyomal occug th factozato of f x, the = f x p x p x p x beg the umbe of tme the eucble polyomal the expoet p x occu the factozato. Th ecompoto alo clealy uque, a calle the pmay ecompoto of f ( x ). Theoem 3.. If B the compao matx of the moc polyomal = λ + λ + + whee p a a B E B C = = E B E = ( cj ).e., c = c = = c = +, +,, + the the vaat polyomal of C ae,,, p. DOI: 0.436/alamt Avace Lea Algeba & Matx Theoy
5 Z. D. Wag et al. Poof. Let B E B C E C c E B ( j ) = λ = = The c = c3 = = c, = c c c, c c c a a mo of oe 3 3 3, c c c,,, = ±.. Thu, D λ = = D λ = D λ = C., By Laplace Theoem, we have that λ λ D = E C = E B = p. Theefoe, the vaat polyomal of C ae The theoem pove.,,, p. λ a the chaactetc The matx C calle the atoal bloc of p polyomal of C pecely the lat vaat polyomal p of C. We ecompoe the vaat polyomal,,, of the λ-matx A to eucble facto ove the umbe fel P = p p p, = p p p, = p p p. Hee, p, p,, p ove P (a wth hghet coeffcet ) that occu a All the powe amog ae all the tct eucble polyomal,,, j j j, j =,,.,,, p p p a fa a they ae tct fom, ae calle the elemetay vo of the λ-matx A the umbe fel P. Theoem 3.. Aume that = S a polyomal of egee, whee p p p S DOI: 0.436/alamt Avace Lea Algeba & Matx Theoy
6 Z. D. Wag et al. p, p,, p ae the tct moc eucble polyomal a,,, ae all potve tege. If C, C,, C ae, epectvely, the atoal bloc of S,,, p p p S the the vaat polyomal of the qua-agoal matx ae,,,. Poof. It eay to ee that C C = C C C = C a the vaat polyomal of C ( =,,, ),,, p by elemetay opeato of λ-matce, C ca be tafome to the caocal fom (ee [8], pp 40-4) C a be futhe tafome to. Thu, = p p =. p o λ-matx, we have that D 3 p λ p λ p λ p λ p λ p p ( 3 ) =,,, = Thu, = = D p p p. = = = = D D D, a the vaat polyomal of The theoem pove. D λ D λ D λ. D λ D λ D λ, =,, =, = DOI: 0.436/alamt Avace Lea Algeba & Matx Theoy
7 Z. D. Wag et al. 4. Qua-Ratoal om of a Matx By combg Joa caocal fom ove the complex fel a the atoal caocal fom ove a umbe fel a ug the atoal bloc of p, we gve the qua-atoal caocal fom of a matx ove a umbe fel. Theoem 4.. If the vaat polyomal of a matx A ove a umbe fel P ae,,,,,,,,, ae the coepog matce of the o-cala vaat polyomal,, Theoem 3., the matx A mla ove the umbe fel P to the qua-agoal matx Poof. It eay to vefy that G =. G = a the vaat polyomal of ( =,,, ) ae,,,. Thu, by elemetay opeato of λ-matce, ca be tafome to the caocal fom a G be futhe tafome to G = =. By techagg ow o colum of G, G ca be tafome to. DOI: 0.436/alamt Avace Lea Algeba & Matx Theoy
8 Z. D. Wag et al. Thu, G a A have ame caocal fom,.e., G a A ae equvalet. Theefoe, A a G ae mla. The theoem pove. The qua-agoal matx G Theoem 4. calle the qua-atoal caocal fom of matx A. Notg that ( =,,, ) Theoem 4. the ect um of the atoal bloc of ome elemetay vo of matx A, we ee that thee lttle bloc matce appeag the qua-atoal caocal fom of A ae pecely the atoal bloc of all elemetay vo of A. Thu, f we f all elemetay vo m,,, p p p m of A a the coepog atoal bloc,,, m, the λe m equvalet to p =. m pm By techagg ow o colum, we ow that equvalet to G G equvalet to Thu, =. DOI: 0.436/alamt Avace Lea Algeba & Matx Theoy
9 Z. D. Wag et al. equvalet to Theefoe, A a the qua-agoal matx ae mla. m Smla to Joa caocal fom of a matx ove the complex fel, f we f all elemetay vo of a matx ove a umbe fel a atoal bloc of thee elemetay vo, the the ect um of thee atoal bloc pecely the qua-atoal caocal fom of the matx. Of coue, the qua-atoal caocal fom of a matx ot uque. But, the qua-atoal caocal fom uque up to a eaagemet of the oe of atoal bloc. 5. Cocluo I th pape, we futhe tuy the atoal caocal fom ove ay umbe fel a gve the qua-atoal caocal fom of a matx by combg Joa a the atoal caocal fom. Ule the compao matce the atoal caocal fom of a matx A [4] [5] [7], thee lttle bloc matce the qua-atoal caocal fom of a matx A ae the atoal bloc of elemetay vo of A a ot the compao matce of the o-cala vaat polyomal of A. Acowlegemet Th wo fue by the laghp Majo Developmet of Jagu Hghe Eucato Ittuto (PPZY05C) a College Stuet Pactce Iovato Tag Pogam ( Y). Refeece [] Rajabalpou, M. (07) A Symmetzato of the Joa Caocal om. Lea. DOI: 0.436/alamt Avace Lea Algeba & Matx Theoy
10 Z. D. Wag et al. Algeba a It Applcato, 58, [] Abo, H., Elu, D., Kahle, T. a Peteo. C. (06) Egecheme a the Joa Caocal om. Lea Algeba a It Applcato, 496, [3] Baoe, M., Lma, J.B. a Campello e Souza, R.M. (06) The Egetuctue a Joa om of the oue Tafom ove el of Chaactetc a a Geealze Vaemoe-Type omula. Lea Algeba a It Applcato, 494, [4] L, S.G. (05) The Stuy o the Popety of Matx Ratoal Caocal om a It Applcato. College Mathematc, 3, (I Chee) [5] Lu, X.Z. (006) A Cotuctve Poof of Extece Theoem fo Ratoal om. College Mathematc,, 5-8. (I Chee) [6] Rajabalpou, M. (03) The Ratoal Caocal om va the Splttg el. Lea Algeba a It Applcato, 439, [7] Hoffma, K. a Kuze, R. (97) Lea Algeba. Eto, Petce-Hall, Ic., Eglewoo Clff. [8] Gatmache,.R. (960) The Theoy of Matce. AMS Chelea Publhg, New Yo. DOI: 0.436/alamt Avace Lea Algeba & Matx Theoy
such that for 1 From the definition of the k-fibonacci numbers, the firsts of them are presented in Table 1. Table 1: First k-fibonacci numbers F 1
Scholas Joual of Egeeg ad Techology (SJET) Sch. J. Eg. Tech. 0; (C):669-67 Scholas Academc ad Scetfc Publshe (A Iteatoal Publshe fo Academc ad Scetfc Resouces) www.saspublshe.com ISSN -X (Ole) ISSN 7-9
More informationUniversity of Pavia, Pavia, Italy. North Andover MA 01845, USA
Iteatoal Joual of Optmzato: heoy, Method ad Applcato 27-5565(Pt) 27-6839(Ole) wwwgph/otma 29 Global Ifomato Publhe (HK) Co, Ltd 29, Vol, No 2, 55-59 η -Peudoleaty ad Effcecy Gogo Gog, Noma G Rueda 2 *
More informationOn EPr Bimatrices II. ON EP BIMATRICES A1 A Hence x. is said to be EP if it satisfies the condition ABx
Iteatoal Joual of Mathematcs ad Statstcs Iveto (IJMSI) E-ISSN: 3 4767 P-ISSN: 3-4759 www.jms.og Volume Issue 5 May. 4 PP-44-5 O EP matces.ramesh, N.baas ssocate Pofesso of Mathematcs, ovt. ts College(utoomous),Kumbakoam.
More informationChapter 2: Descriptive Statistics
Chapte : Decptve Stattc Peequte: Chapte. Revew of Uvaate Stattc The cetal teecy of a oe o le yetc tbuto of a et of teval, o hghe, cale coe, ofte uaze by the athetc ea, whch efe a We ca ue the ea to ceate
More informationHarmonic Curvatures in Lorentzian Space
BULLETIN of the Bull Malaya Math Sc Soc Secod See 7-79 MALAYSIAN MATEMATICAL SCIENCES SOCIETY amoc Cuvatue Loetza Space NEJAT EKMEKÇI ILMI ACISALIOĞLU AND KĀZIM İLARSLAN Aaa Uvety Faculty of Scece Depatmet
More informationDistribution of Geometrically Weighted Sum of Bernoulli Random Variables
Appled Mathematc, 0,, 8-86 do:046/am095 Publhed Ole Novembe 0 (http://wwwscrpog/oual/am) Dtbuto of Geometcally Weghted Sum of Beoull Radom Vaable Abtact Deepeh Bhat, Phazamle Kgo, Ragaath Naayaachaya Ratthall
More information= y and Normed Linear Spaces
304-50 LINER SYSTEMS Lectue 8: Solutos to = ad Nomed Lea Spaces 73 Fdg N To fd N, we eed to chaacteze all solutos to = 0 Recall that ow opeatos peseve N, so that = 0 = 0 We ca solve = 0 ecusvel backwads
More information( m is the length of columns of A ) spanned by the columns of A : . Select those columns of B that contain a pivot; say those are Bi
Assgmet /MATH 47/Wte Due: Thusday Jauay The poblems to solve ae umbeed [] to [] below Fst some explaatoy otes Fdg a bass of the colum-space of a max ad povg that the colum ak (dmeso of the colum space)
More informationXII. Addition of many identical spins
XII. Addto of may detcal sps XII.. ymmetc goup ymmetc goup s the goup of all possble pemutatos of obects. I total! elemets cludg detty opeato. Each pemutato s a poduct of a ceta fte umbe of pawse taspostos.
More informationRECAPITULATION & CONDITIONAL PROBABILITY. Number of favourable events n E Total number of elementary events n S
Fomulae Fo u Pobablty By OP Gupta [Ida Awad We, +91-9650 350 480] Impotat Tems, Deftos & Fomulae 01 Bascs Of Pobablty: Let S ad E be the sample space ad a evet a expemet espectvely Numbe of favouable evets
More informationTrace of Positive Integer Power of Adjacency Matrix
Global Joual of Pue ad Appled Mathematcs. IN 097-78 Volume, Numbe 07), pp. 079-087 Reseach Ida Publcatos http://www.publcato.com Tace of Postve Itege Powe of Adacecy Matx Jagdsh Kuma Pahade * ad Mao Jha
More informationThe Linear Probability Density Function of Continuous Random Variables in the Real Number Field and Its Existence Proof
MATEC Web of Cofeeces ICIEA 06 600 (06) DOI: 0.05/mateccof/0668600 The ea Pobablty Desty Fucto of Cotuous Radom Vaables the Real Numbe Feld ad Its Estece Poof Yya Che ad Ye Collee of Softwae, Taj Uvesty,
More informationThe Geometric Proof of the Hecke Conjecture
The Geometc Poof of the Hecke Cojectue Kada Sh Depatmet of Mathematc Zhejag Ocea Uvety Zhouha Cty 6 Zhejag Povce Cha Atact Begg fom the eoluto of Dchlet fucto ug the e poduct fomula of two fte-dmeoal vecto
More informationProfessor Wei Zhu. 1. Sampling from the Normal Population
AMS570 Pofesso We Zhu. Samplg fom the Nomal Populato *Example: We wsh to estmate the dstbuto of heghts of adult US male. It s beleved that the heght of adult US male follows a omal dstbuto N(, ) Def. Smple
More informationInequalities for Dual Orlicz Mixed Quermassintegrals.
Advaces Pue Mathematcs 206 6 894-902 http://wwwscpog/joual/apm IN Ole: 260-0384 IN Pt: 260-0368 Iequaltes fo Dual Olcz Mxed Quemasstegals jua u chool of Mathematcs ad Computatoal cece Hua Uvesty of cece
More informationIntuitionistic Fuzzy Stability of n-dimensional Cubic Functional Equation: Direct and Fixed Point Methods
Ite. J. Fuzzy Mathematcal Achve Vol. 7 No. 205 - ISSN: 220 242 (P 220 250 (ole Publhed o2 Jauay 205 www.eeachmathc.og Iteatoal Joual of Itutotc Fuzzy Stablty of -Dmeoal Cubc Fuctoal Equato: Dect ad Fxed
More informationFairing of Parametric Quintic Splines
ISSN 46-69 Eglad UK Joual of Ifomato ad omputg Scece Vol No 6 pp -8 Fag of Paametc Qutc Sples Yuau Wag Shagha Isttute of Spots Shagha 48 ha School of Mathematcal Scece Fuda Uvesty Shagha 4 ha { P t )}
More informationThen the number of elements of S of weight n is exactly the number of compositions of n into k parts.
Geneating Function In a geneal combinatoial poblem, we have a univee S of object, and we want to count the numbe of object with a cetain popety. Fo example, if S i the et of all gaph, we might want to
More informationSYSTEMS OF NON-LINEAR EQUATIONS. Introduction Graphical Methods Close Methods Open Methods Polynomial Roots System of Multivariable Equations
SYSTEMS OF NON-LINEAR EQUATIONS Itoduto Gaphal Method Cloe Method Ope Method Polomal Root Stem o Multvaale Equato Chapte Stem o No-Lea Equato /. Itoduto Polem volvg o-lea equato egeeg lude optmato olvg
More informationMinimum Hyper-Wiener Index of Molecular Graph and Some Results on Szeged Related Index
Joual of Multdscplay Egeeg Scece ad Techology (JMEST) ISSN: 359-0040 Vol Issue, Febuay - 05 Mmum Hype-Wee Idex of Molecula Gaph ad Some Results o eged Related Idex We Gao School of Ifomato Scece ad Techology,
More informationRandomly Weighted Averages on Order Statistics
Apple Mathematcs 3 4 34-346 http://oog/436/am3498 Publshe Ole Septembe 3 (http://wwwscpog/joual/am Raomly Weghte Aveages o Oe Statstcs Home Haj Hasaaeh Lela Ma Ghasem Depatmet of Statstcs aculty of Mathematcal
More informationBy the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences
Chapte : Theoy of Modula Aithmetic 8 Sectio D Chiese Remaide Theoem By the ed of this sectio you will be able to pove the Chiese Remaide Theoem apply this theoem to solve simultaeous liea cogueces The
More informationSolving the Dirac Equation: Using Fourier Transform
McNa Schola Reeach Jounal Volume Atcle Solvng the ac quaton: Ung oue Tanfom Vncent P. Bell mby-rddle Aeonautcal Unvety, Vncent.Bell@my.eau.edu ollow th and addtonal wok at: http://common.eau.edu/na Recommended
More informationIterative Algorithm for a Split Equilibrium Problem and Fixed Problem for Finite Asymptotically Nonexpansive Mappings in Hilbert Space
Flomat 31:5 (017), 143 1434 DOI 10.98/FIL170543W Publshed by Faculty of Sceces ad Mathematcs, Uvesty of Nš, Seba Avalable at: http://www.pmf..ac.s/flomat Iteatve Algothm fo a Splt Equlbum Poblem ad Fxed
More informationANOTHER INTEGER NUMBER ALGORITHM TO SOLVE LINEAR EQUATIONS (USING CONGRUENCY)
ANOTHER INTEGER NUMBER ALGORITHM TO SOLVE LINEAR EQUATIONS (USING CONGRUENCY) Floet Smdche, Ph D Aocte Pofeo Ch of Deptmet of Mth & Scece Uvety of New Mexco 2 College Rod Gllup, NM 873, USA E-ml: md@um.edu
More informationSOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES
#A17 INTEGERS 9 2009), 181-190 SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES Deick M. Keiste Depatmet of Mathematics, Pe State Uivesity, Uivesity Pak, PA 16802 dmk5075@psu.edu James A. Selles Depatmet
More informationUNIT 7 RANK CORRELATION
UNIT 7 RANK CORRELATION Rak Correlato Structure 7. Itroucto Objectves 7. Cocept of Rak Correlato 7.3 Dervato of Rak Correlato Coeffcet Formula 7.4 Te or Repeate Raks 7.5 Cocurret Devato 7.6 Summar 7.7
More informationRANDOM SYSTEMS WITH COMPLETE CONNECTIONS AND THE GAUSS PROBLEM FOR THE REGULAR CONTINUED FRACTIONS
RNDOM SYSTEMS WTH COMPETE CONNECTONS ND THE GUSS PROBEM FOR THE REGUR CONTNUED FRCTONS BSTRCT Da ascu o Coltescu Naval cademy Mcea cel Bata Costata lascuda@gmalcom coltescu@yahoocom Ths pape peset the
More informationASYMPTOTICS OF THE GENERALIZED STATISTICS FOR TESTING THE HYPOTHESIS UNDER RANDOM CENSORING
IJRRAS 3 () Novembe www.apape.com/volume/vol3iue/ijrras_3.pdf ASYMPOICS OF HE GENERALIZE SAISICS FOR ESING HE HYPOHESIS UNER RANOM CENSORING A.A. Abduhukuov & N.S. Numuhamedova Natoal Uvety of Uzbekta
More informationThe calculation of the characteristic and non-characteristic harmonic current of the rectifying system
The calculato of the chaactestc a o-chaactestc hamoc cuet of the ectfyg system Zhag Ruhua, u Shagag, a Luguag, u Zhegguo The sttute of Electcal Egeeg, Chese Acaemy of Sceces, ejg, 00080, Cha. Zhag Ruhua,
More informationTHE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 9, Number 3/2008, pp
THE PUBLISHIN HOUSE PROCEEDINS OF THE ROMANIAN ACADEMY, Seres A, OF THE ROMANIAN ACADEMY Volume 9, Number 3/8, THE UNITS IN Stela Corelu ANDRONESCU Uversty of Pteşt, Deartmet of Mathematcs, Târgu Vale
More informationQuestion 1. Typical Cellular System. Some geometry TELE4353. About cellular system. About cellular system (2)
TELE4353 Moble a atellte Commucato ystems Tutoal 1 (week 3-4 4 Questo 1 ove that fo a hexagoal geomety, the co-chael euse ato s gve by: Q (3 N Whee N + j + j 1/ 1 Typcal Cellula ystem j cells up cells
More informationThe Pigeonhole Principle 3.4 Binomial Coefficients
Discete M athematic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Ageda Ch 3. The Pigeohole Piciple
More informationCh 3.4 Binomial Coefficients. Pascal's Identit y and Triangle. Chapter 3.2 & 3.4. South China University of Technology
Disc ete Mathem atic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Pigeohole Piciple Suppose that a
More informationProgression. CATsyllabus.com. CATsyllabus.com. Sequence & Series. Arithmetic Progression (A.P.) n th term of an A.P.
Pogessio Sequece & Seies A set of umbes whose domai is a eal umbe is called a SEQUENCE ad sum of the sequece is called a SERIES. If a, a, a, a 4,., a, is a sequece, the the expessio a + a + a + a 4 + a
More informationSpectral Problems of Two-Parameter System of Operators
Pue ad Appled Matheatc Joual 5; 4(4-: 33-37 Publhed ole Augut, 5 (http://wwwcecepublhggoupco//pa do: 648/pa5447 ISSN: 36-979 (Pt; ISSN: 36-98 (Ole Spectal Poble of Two-Paaete Syte of Opeato Rahhada Dhabaadeh
More informationLecture 6: October 16, 2017
Ifomatio ad Codig Theoy Autum 207 Lectue: Madhu Tulsiai Lectue 6: Octobe 6, 207 The Method of Types Fo this lectue, we will take U to be a fiite uivese U, ad use x (x, x 2,..., x to deote a sequece of
More informationA NOTE ON DOMINATION PARAMETERS IN RANDOM GRAPHS
Discussioes Mathematicae Gaph Theoy 28 (2008 335 343 A NOTE ON DOMINATION PARAMETERS IN RANDOM GRAPHS Athoy Boato Depatmet of Mathematics Wilfid Lauie Uivesity Wateloo, ON, Caada, N2L 3C5 e-mail: aboato@oges.com
More informationChapter 3: Theory of Modular Arithmetic 38
Chapte 3: Theoy of Modula Aithmetic 38 Section D Chinese Remainde Theoem By the end of this section you will be able to pove the Chinese Remainde Theoem apply this theoem to solve simultaneous linea conguences
More informationBINOMIAL THEOREM An expression consisting of two terms, connected by + or sign is called a
BINOMIAL THEOREM hapte 8 8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4 5y, etc., ae all biomial epessios. 8.. Biomial theoem If
More informationRange Symmetric Matrices in Minkowski Space
BULLETIN of the Bull. alaysia ath. Sc. Soc. (Secod Seies) 3 (000) 45-5 LYSIN THETICL SCIENCES SOCIETY Rae Symmetic atices i ikowski Space.R. EENKSHI Depatmet of athematics, amalai Uivesity, amalaiaa 608
More informationBINOMIAL THEOREM NCERT An expression consisting of two terms, connected by + or sign is called a
8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4, etc., ae all biomial 5y epessios. 8.. Biomial theoem BINOMIAL THEOREM If a ad b ae
More informationSome Integrals Pertaining Biorthogonal Polynomials and Certain Product of Special Functions
Global Joual o Scece Fote Reeach atheatc ad Deco Scece Volue Iue Veo Te : Double Bld ee Reewed Iteatoal Reeach Joual ublhe: Global Joual Ic SA Ole ISSN: 49-466 & t ISSN: 975-5896 Soe Itegal etag Bothogoal
More informationON THE CONVERGENCE THEOREMS OF THE McSHANE INTEGRAL FOR RIESZ-SPACES-VALUED FUNCTIONS DEFINED ON REAL LINE
O The Covegece Theoems... (Muslm Aso) ON THE CONVERGENCE THEOREMS OF THE McSHANE INTEGRAL FOR RIESZ-SPACES-VALUED FUNCTIONS DEFINED ON REAL LINE Muslm Aso, Yosephus D. Sumato, Nov Rustaa Dew 3 ) Mathematcs
More informationSome Integral Mean Estimates for Polynomials
Iteatioal Mathematical Foum, Vol. 8, 23, o., 5-5 HIKARI Ltd, www.m-hikai.com Some Itegal Mea Estimates fo Polyomials Abdullah Mi, Bilal Ahmad Da ad Q. M. Dawood Depatmet of Mathematics, Uivesity of Kashmi
More informationOn Eigenvalues of Nonlinear Operator Pencils with Many Parameters
Ope Scece Joual of Matheatc ad Applcato 5; 3(4): 96- Publhed ole Jue 5 (http://wwwopececeoleco/oual/oa) O Egevalue of Nolea Opeato Pecl wth May Paaete Rakhhada Dhabaadeh Guay Salaova Depatet of Fuctoal
More informationSeveral new identities involving Euler and Bernoulli polynomials
Bull. Math. Soc. Sci. Math. Roumanie Tome 9107 No. 1, 016, 101 108 Seveal new identitie involving Eule and Benoulli polynomial by Wang Xiaoying and Zhang Wenpeng Abtact The main pupoe of thi pape i uing
More informationFIXED POINT AND HYERS-ULAM-RASSIAS STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN BANACH SPACES
IJRRAS 6 () July 0 www.apapess.com/volumes/vol6issue/ijrras_6.pdf FIXED POINT AND HYERS-UAM-RASSIAS STABIITY OF A QUADRATIC FUNCTIONA EQUATION IN BANACH SPACES E. Movahedia Behbaha Khatam Al-Abia Uivesity
More informationELEMENTS OF NUMBER THEORY. In the following we will use mainly integers and positive integers. - the set of integers - the set of positive integers
ELEMENTS OF NUMBER THEORY I the followg we wll use aly tegers a ostve tegers Ζ = { ± ± ± K} - the set of tegers Ν = { K} - the set of ostve tegers Oeratos o tegers: Ato Each two tegers (ostve tegers) ay
More informationLecture 24: Observability and Constructibility
ectue 24: Obsevability ad Costuctibility 7 Obsevability ad Costuctibility Motivatio: State feedback laws deped o a kowledge of the cuet state. I some systems, xt () ca be measued diectly, e.g., positio
More informationA Bijective Approach to the Permutational Power of a Priority Queue
A Bijective Appoach to the Pemutational Powe of a Pioity Queue Ia M. Gessel Kuang-Yeh Wang Depatment of Mathematics Bandeis Univesity Waltham, MA 02254-9110 Abstact A pioity queue tansfoms an input pemutation
More informationDivisibility. c = bf = (ae)f = a(ef) EXAMPLE: Since 7 56 and , the Theorem above tells us that
Divisibility DEFINITION: If a and b ae integes with a 0, we say that a divides b if thee is an intege c such that b = ac. If a divides b, we also say that a is a diviso o facto of b. NOTATION: d n means
More informationNew problems in universal algebraic geometry illustrated by boolean equations
New poblems in univesal algebaic geomety illustated by boolean equations axiv:1611.00152v2 [math.ra] 25 Nov 2016 Atem N. Shevlyakov Novembe 28, 2016 Abstact We discuss new poblems in univesal algebaic
More informationLower Bounds for Cover-Free Families
Loe Bouds fo Cove-Fee Families Ali Z. Abdi Covet of Nazaeth High School Gade, Abas 7, Haifa Nade H. Bshouty Dept. of Compute Sciece Techio, Haifa, 3000 Apil, 05 Abstact Let F be a set of blocks of a t-set
More informationMapping Radius of Regular Function and Center of Convex Region. Duan Wenxi
d Iteatioal Cofeece o Electical Compute Egieeig ad Electoics (ICECEE 5 Mappig adius of egula Fuctio ad Cete of Covex egio Dua Wexi School of Applied Mathematics Beijig Nomal Uivesity Zhuhai Chia 363463@qqcom
More informationSums of Involving the Harmonic Numbers and the Binomial Coefficients
Ameica Joual of Computatioal Mathematics 5 5 96-5 Published Olie Jue 5 i SciRes. http://www.scip.og/oual/acm http://dx.doi.og/.46/acm.5.58 Sums of Ivolvig the amoic Numbes ad the Biomial Coefficiets Wuyugaowa
More informationTHE CONE THEOREM JOEL A. TROPP. Abstract. We prove a fixed point theorem for functions which are positive with respect to a cone in a Banach space.
THE ONE THEOEM JOEL A. TOPP Abstact. We pove a fixed point theoem fo functions which ae positive with espect to a cone in a Banach space. 1. Definitions Definition 1. Let X be a eal Banach space. A subset
More informationOn ARMA(1,q) models with bounded and periodically correlated solutions
Reseach Repot HSC/03/3 O ARMA(,q) models with bouded ad peiodically coelated solutios Aleksade Weo,2 ad Agieszka Wy oma ska,2 Hugo Steihaus Cete, Woc aw Uivesity of Techology 2 Istitute of Mathematics,
More informationa) The average (mean) of the two fractions is halfway between them: b) The answer is yes. Assume without loss of generality that p < r.
Solutios to MAML Olympiad Level 00. Factioated a) The aveage (mea) of the two factios is halfway betwee them: p ps+ q ps+ q + q s qs qs b) The aswe is yes. Assume without loss of geeality that p
More informationNONDIFFERENTIABLE MATHEMATICAL PROGRAMS. OPTIMALITY AND HIGHER-ORDER DUALITY RESULTS
HE PUBLISHING HOUSE PROCEEDINGS OF HE ROMANIAN ACADEMY, See A, OF HE ROMANIAN ACADEMY Volue 9, Nube 3/8,. NONDIFFERENIABLE MAHEMAICAL PROGRAMS. OPIMALIY AND HIGHER-ORDER DUALIY RESULS Vale PREDA Uvety
More informationOn a Problem of Littlewood
Ž. JOURAL OF MATHEMATICAL AALYSIS AD APPLICATIOS 199, 403 408 1996 ARTICLE O. 0149 O a Poblem of Littlewood Host Alze Mosbache Stasse 10, 51545 Waldbol, Gemay Submitted by J. L. Bee Received May 19, 1995
More informationOn the Quasi-Hyperbolic Kac-Moody Algebra QHA7 (2)
Ieaoal Reeach Joual of Egeeg ad Techology (IRJET) e-issn: 9 - Volume: Iue: May- www.e.e -ISSN: 9-7 O he Qua-Hyebolc Kac-Moody lgeba QH7 () Uma Mahewa., Khave. S Deame of Mahemac Quad-E-Mllah Goveme College
More informationComplementary Dual Subfield Linear Codes Over Finite Fields
1 Complemetay Dual Subfield Liea Codes Ove Fiite Fields Kiagai Booiyoma ad Somphog Jitma,1 Depatmet of Mathematics, Faculty of Sciece, Silpao Uivesity, Naho Pathom 73000, hailad e-mail : ai_b_555@hotmail.com
More informationAn Indian Journal FULL PAPER ABSTRACT KEYWORDS. Trade Science Inc. Super-efficiency infeasibility and zero data in DEA: An alternative approach
[Type text] [Type text] [Type text] ISSN : 0974-7435 Volue 0 Iue 7 BoTechology 204 A Ida Joual FULL PAPER BTAIJ, 0(7), 204 [773-779] Supe-effcecy feablty ad zeo data DEA: A alteatve appoach Wag Q, Guo
More informationON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS
Joual of Pue ad Alied Mathematics: Advaces ad Alicatios Volume 0 Numbe 03 Pages 5-58 ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS ALI H HAKAMI Deatmet of Mathematics
More informationConditional Convergence of Infinite Products
Coditioal Covegece of Ifiite Poducts William F. Tech Ameica Mathematical Mothly 106 1999), 646-651 I this aticle we evisit the classical subject of ifiite poducts. Fo stadad defiitios ad theoems o this
More informationExponential Generating Functions - J. T. Butler
Epoetal Geeatg Fuctos - J. T. Butle Epoetal Geeatg Fuctos Geeatg fuctos fo pemutatos. Defto: a +a +a 2 2 + a + s the oday geeatg fucto fo the sequece of teges (a, a, a 2, a, ). Ep. Ge. Fuc.- J. T. Butle
More informationChapter 7 Varying Probability Sampling
Chapte 7 Vayg Pobablty Samplg The smple adom samplg scheme povdes a adom sample whee evey ut the populato has equal pobablty of selecto. Ude ceta ccumstaces, moe effcet estmatos ae obtaed by assgg uequal
More informationAPPLICATIONS OF SEMIGENERALIZED -CLOSED SETS
Intenatonal Jounal of Mathematcal Engneeng Scence ISSN : 22776982 Volume Issue 4 (Apl 202) http://www.mes.com/ https://stes.google.com/ste/mesounal/ APPLICATIONS OF SEMIGENERALIZED CLOSED SETS G.SHANMUGAM,
More informationSensorless A.C. Drive with Vector Controlled Synchronous Motor
Seole A.C. Dve wth Vecto Cotolle Sychoo Moto Ořej Fše VŠB-echcal Uvety of Otava, Faclty of Electcal Egeeg a Ifomatc, Deatmet of Powe Electoc a Electcal Dve, 17.ltoa 15, 78 33 Otava-Poba, Czech eblc oej.fe@vb.cz
More informationof the contestants play as Falco, and 1 6
JHMT 05 Algeba Test Solutions 4 Febuay 05. In a Supe Smash Bothes tounament, of the contestants play as Fox, 3 of the contestants play as Falco, and 6 of the contestants play as Peach. Given that thee
More information2012 GCE A Level H2 Maths Solution Paper Let x,
GCE A Level H Maths Solutio Pape. Let, y ad z be the cost of a ticet fo ude yeas, betwee ad 5 yeas, ad ove 5 yeas categoies espectively. 9 + y + 4z =. 7 + 5y + z = 8. + 4y + 5z = 58.5 Fo ude, ticet costs
More informationSolution to HW 3, Ma 1a Fall 2016
Solution to HW 3, Ma a Fall 206 Section 2. Execise 2: Let C be a subset of the eal numbes consisting of those eal numbes x having the popety that evey digit in the decimal expansion of x is, 3, 5, o 7.
More informationχ be any function of X and Y then
We have show that whe we ae gve Y g(), the [ ] [ g() ] g() f () Y o all g ()() f d fo dscete case Ths ca be eteded to clude fuctos of ay ube of ado vaables. Fo eaple, suppose ad Y ae.v. wth jot desty fucto,
More informationInterior and Exterior Differential Systems for Lie Algebroids
Aaces Pue Mathematcs 0 45-49 o:0436/am05044 Publshe Ole Setembe 0 (htt://wwwscrpog/oual/am) Iteo a Eteo Dffeetal Systems fo Le Algebos Costat M Acuş Secoay School CORELIUS RADU Răeşt Vllage Go Couty Româa
More informationMethod for Approximating Irrational Numbers
Method fo Appoximating Iational Numbes Eic Reichwein Depatment of Physics Univesity of Califonia, Santa Cuz June 6, 0 Abstact I will put foth an algoithm fo poducing inceasingly accuate ational appoximations
More information9.1 The multiplicative group of a finite field. Theorem 9.1. The multiplicative group F of a finite field is cyclic.
Chapte 9 Pimitive Roots 9.1 The multiplicative goup of a finite fld Theoem 9.1. The multiplicative goup F of a finite fld is cyclic. Remak: In paticula, if p is a pime then (Z/p) is cyclic. In fact, this
More informationGeneralized Fibonacci-Lucas Sequence
Tuish Joual of Aalysis ad Numbe Theoy, 4, Vol, No 6, -7 Available olie at http://pubssciepubcom/tjat//6/ Sciece ad Educatio Publishig DOI:6/tjat--6- Geealized Fiboacci-Lucas Sequece Bijeda Sigh, Ompaash
More informationFULLY RIGHT PURE GROUP RINGS (Gelanggang Kumpulan Tulen Kanan Penuh)
Joual of Qualty Measuemet ad Aalyss JQMA 3(), 07, 5-34 Jual Pegukua Kualt da Aalss FULLY IGHT PUE GOUP INGS (Gelaggag Kumpula Tule Kaa Peuh) MIKHLED ALSAAHEAD & MOHAMED KHEI AHMAD ABSTACT I ths pape, we
More informationStratification Analysis of Certain Nakayama Algebras
Advaces ue athematcs, 5, 5, 85-855 ublshed Ole Decembe 5 ScRes http://wwwscpog/joual/apm http://dxdoog/46/apm55479 Statfcato Aalyss of Ceta Nakayama Algebas José Fdel Heádez Advícula, Rafael Facsco Ochoa
More informationCouncil for Innovative Research
Geometc-athmetc Idex ad Zageb Idces of Ceta Specal Molecula Gaphs efe X, e Gao School of Tousm ad Geogaphc Sceces, Yua Nomal Uesty Kumg 650500, Cha School of Ifomato Scece ad Techology, Yua Nomal Uesty
More informationResearch Article On Alzer and Qiu s Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function
Abstact and Applied Analysis Volume 011, Aticle ID 697547, 7 pages doi:10.1155/011/697547 Reseach Aticle On Alze and Qiu s Conjectue fo Complete Elliptic Integal and Invese Hypebolic Tangent Function Yu-Ming
More informationVECTOR MECHANICS FOR ENGINEERS: Vector Mechanics for Engineers: Dynamics. In the current chapter, you will study the motion of systems of particles.
Seeth Edto CHPTER 4 VECTOR MECHNICS FOR ENINEERS: DYNMICS Fedad P. ee E. Russell Johsto, J. Systems of Patcles Lectue Notes: J. Walt Ole Texas Tech Uesty 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed. Seeth
More information< 1. max x B(0,1) f. ν ds(y) Use Poisson s formula for the ball to prove. (r x ) x y n ds(y) (x B0 (0, r)). 1 nα(n)r n 1
7 On the othe hand, u x solves { u n in U u on U, so which implies that x G(x, y) x n ng(x, y) < n (x B(, )). Theefoe u(x) fo all x B(, ). G ν ds(y) + max g G x ν ds(y) + C( max g + max f ) f(y)g(x, y)
More informationChapter Linear Regression
Chpte 6.3 Le Regesso Afte edg ths chpte, ou should be ble to. defe egesso,. use sevel mmzg of esdul cte to choose the ght cteo, 3. deve the costts of le egesso model bsed o lest sques method cteo,. use
More informationMATH /19: problems for supervision in week 08 SOLUTIONS
MATH10101 2018/19: poblems fo supevisio i week 08 Q1. Let A be a set. SOLUTIONS (i Pove that the fuctio c: P(A P(A, defied by c(x A \ X, is bijective. (ii Let ow A be fiite, A. Use (i to show that fo each
More informationMaximum Walk Entropy Implies Walk Regularity
Maxmum Walk Etropy Imples Walk Regularty Eresto Estraa, a José. e la Peña Departmet of Mathematcs a Statstcs, Uversty of Strathclye, Glasgow G XH, U.K., CIMT, Guaajuato, Mexco BSTRCT: The oto of walk etropy
More informationCounting Functions and Subsets
CHAPTER 1 Coutig Fuctios ad Subsets This chapte of the otes is based o Chapte 12 of PJE See PJE p144 Hee ad below, the efeeces to the PJEccles book ae give as PJE The goal of this shot chapte is to itoduce
More informationTrignometric Inequations and Fuzzy Information Theory
Iteratoal Joural of Scetfc ad Iovatve Mathematcal Reearch (IJSIMR) Volume, Iue, Jauary - 0, PP 00-07 ISSN 7-07X (Prt) & ISSN 7- (Ole) www.arcjoural.org Trgometrc Iequato ad Fuzzy Iformato Theory P.K. Sharma,
More informationFIBONACCI-LIKE SEQUENCE ASSOCIATED WITH K-PELL, K-PELL-LUCAS AND MODIFIED K-PELL SEQUENCES
Joual of Appled Matheatcs ad Coputatoal Mechacs 7, 6(), 59-7 www.ac.pcz.pl p-issn 99-9965 DOI:.75/jac.7..3 e-issn 353-588 FIBONACCI-LIKE SEQUENCE ASSOCIATED WITH K-PELL, K-PELL-LUCAS AND MODIFIED K-PELL
More informationConsumer theory. A. The preference ordering B. The feasible set C. The consumption decision. A. The preference ordering. Consumption bundle
Thomas Soesso Mcoecoomcs Lecte Cosme theoy A. The efeece odeg B. The feasble set C. The cosmto decso A. The efeece odeg Cosmto bdle x ( 2 x, x,... x ) x Assmtos: Comleteess 2 Tastvty 3 Reflexvty 4 No-satato
More informationA proof of the binomial theorem
A poof of the binomial theoem If n is a natual numbe, let n! denote the poduct of the numbes,2,3,,n. So! =, 2! = 2 = 2, 3! = 2 3 = 6, 4! = 2 3 4 = 24 and so on. We also let 0! =. If n is a non-negative
More informationTaylor Transformations into G 2
Iteatioal Mathematical Foum, 5,, o. 43, - 3 Taylo Tasfomatios ito Mulatu Lemma Savaah State Uivesity Savaah, a 344, USA Lemmam@savstate.edu Abstact. Though out this pape, we assume that
More informationONE-POINT CODES USING PLACES OF HIGHER DEGREE
ONE-POINT CODES USING PLACES OF HIGHER DEGREE GRETCHEN L. MATTHEWS AND TODD W. MICHEL DEPARTMENT OF MATHEMATICAL SCIENCES CLEMSON UNIVERSITY CLEMSON, SC 29634-0975 U.S.A. E-MAIL: GMATTHE@CLEMSON.EDU, TMICHEL@CLEMSON.EDU
More informationThird handout: On the Gini Index
Thrd hadout: O the dex Corrado, a tala statstca, proposed (, 9, 96) to measure absolute equalt va the mea dfferece whch s defed as ( / ) where refers to the total umber of dvduals socet. Assume that. The
More informationarxiv: v1 [math.co] 6 Mar 2008
An uppe bound fo the numbe of pefect matchings in gaphs Shmuel Fiedland axiv:0803.0864v [math.co] 6 Ma 2008 Depatment of Mathematics, Statistics, and Compute Science, Univesity of Illinois at Chicago Chicago,
More informationGreatest term (numerically) in the expansion of (1 + x) Method 1 Let T
BINOMIAL THEOREM_SYNOPSIS Geatest tem (umeically) i the epasio of ( + ) Method Let T ( The th tem) be the geatest tem. Fid T, T, T fom the give epasio. Put T T T ad. Th will give a iequality fom whee value
More informationOn the Explicit Determinants and Singularities of r-circulant and Left r-circulant Matrices with Some Famous Numbers
O the Explicit Detemiats Sigulaities of -ciculat Left -ciculat Matices with Some Famous Numbes ZHAOLIN JIANG Depatmet of Mathematics Liyi Uivesity Shuaglig Road Liyi city CHINA jzh08@siacom JUAN LI Depatmet
More informationGeneralizations and analogues of the Nesbitt s inequality
OCTOGON MATHEMATICAL MAGAZINE Vol 17, No1, Apil 2009, pp 215-220 ISSN 1222-5657, ISBN 978-973-88255-5-0, wwwhetfaluo/octogo 215 Geealiatios ad aalogues of the Nesbitt s iequalit Fuhua Wei ad Shahe Wu 19
More informationJohns Hopkins University Department of Biostatistics Math Review for Introductory Courses
Johs Hopks Uverst Departmet of Bostatstcs Math Revew for Itroductor Courses Ratoale Bostatstcs courses wll rel o some fudametal mathematcal relatoshps, fuctos ad otato. The purpose of ths Math Revew s
More information